NO

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(+ 1, 2)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(s(x),y) -> s(+(x,y))
+(0,0) -> 0
+(x,s(y)) -> s(+(y,x))
s(s(x)) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x y)
(REPLACEMENT-MAP
(+ 1, 2)
(s 1)
(0)
(fSNonEmpty)
)
(RULES
+(s(x),y) -> s(+(x,y))
+(0,0) -> 0
+(x,s(y)) -> s(+(y,x))
s(s(x)) -> x
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 +(s(x),y) -> s(+(x,y))
 +(0,0) -> 0
 +(x,s(y)) -> s(+(y,x))
 s(s(x)) -> x
-> Vars:
 x, y, x, y, x

-> Rlps:
 (rule: +(s(x),y) -> s(+(x,y)), id: 1, possubterms: +(s(x),y)->[], s(x)->[1])
 (rule: +(0,0) -> 0, id: 2, possubterms: +(0,0)->[], 0->[1], 0->[2])
 (rule: +(x,s(y)) -> s(+(y,x)), id: 3, possubterms: +(x,s(y))->[], s(y)->[2])
 (rule: s(s(x)) -> x, id: 4, possubterms: s(s(x))->[], s(x)->[1])

-> Unifications:
(R1 unifies with R4 at p: [1], l: +(s(x),y), lp: s(x), sig: {x -> s(x')}, l': s(s(x')), r: s(+(x,y)), r': x')
(R3 unifies with R1 at p: [], l: +(x,s(y)), lp: +(x,s(y)), sig: {x -> s(x'),y' -> s(y)}, l': +(s(x'),y'), r: s(+(y,x)), r': s(+(x',y')))
(R3 unifies with R4 at p: [2], l: +(x,s(y)), lp: s(y), sig: {y -> s(x')}, l': s(s(x')), r: s(+(y,x)), r': x')
(R4 unifies with R4 at p: [1], l: s(s(x)), lp: s(x), sig: {x -> s(x')}, l': s(s(x')), r: x, r': x')

-> Critical pairs info:
<s(x'),s(x')>   =>  Trivial, Not overlay, Proper, NW0, N1
<+(x,x'),s(+(s(x'),x))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<s(+(x',s(y))),s(+(y,s(x')))>   =>  Not trivial, Overlay, Proper, NW0, N3
<+(x',y),s(+(s(x'),y))>   =>  Not trivial, Not overlay, Proper, NW0, N4

-> Problem conclusions:
 Left linear, Right linear, Linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a TRS 


Problem 1: 
No Convergence Brute Force Procedure:
-> Rewritings:
s: +(x,x')
Nodes: [0]
Edges: []
ID: 0 => ('+(x,x')', D0)
t: s(+(s(x'),x))
Nodes: [0,1,2]
Edges: [(0,1),(1,2)]
ID: 0 => ('s(+(s(x'),x))', D0)
ID: 1 => ('s(s(+(x',x)))', D1, R1, P[1], S{x5 -> x', x6 -> x}), NR: 's(+(x',x))'
ID: 2 => ('+(x',x)', D2, R4, P[], S{x9 -> +(x',x)}), NR: '+(x',x)'
+(x,x') ->* no union *<- s(+(s(x'),x))
"Not joinable"

The problem is not confluent.